Caption: The fitting of observed supernova light curves to a fiducial normal SN Ia light curve illustrated dynamically by NAAP applet: Supernova Light Curve Fitting Explorer provided by NAAP.
We derive the procedure below.
where F is flux (energy per unit time per unit area perpendicular to the flux direction), L is luminosity (energy per unit time output by a spherically symmetric source), and r is the distance to the source assuming a STATIC SYSTEM.
We assume no extinction due to interstellar dust in our discussion---but you have to worry about it in real cases.
We invert the flux formula to get the formula:
If their L is known, the luminosity distance formula can be used to get the luminosity distances.
SNe Ia are NOT exactly standard candles.
However, SNe Ia are approximately standard candles for crude results and they can be corrected to be very nearly standard candles for much better results.
Here we will just assume SNe Ia are standard candles as a heuristic simplification.
Thus, we can find the luminosity distances to SNe Ia.
At present, the record most distant SNe Ia is at cosmological redshift z ≅ 2.25 (see Wikipedia: List of most distant supernovae, Rodney et al. 2015) corresponding to luminosity distance ≅ 17 Gpc, proper distance ≅ 7 Gpc, and lookback time ≅ 10 Gyr.
It's NOT clear that SNe Ia at z ≅ 2 can be used to determine accurate luminosity distances yet.
The luminosity distances are NOT proper distances for the two following reasons:
Astronomical objects in the cosmological realm move substantially between emitting a light signal and the light being observed at Earth. Space literally grows under the light signal as travels to Earth.
Proper distances are NOT direct observables. We can only know what they are for astronomical objects for a given cosmologicaly model.
At present, the overwhelmingly favored cosmological model is the Λ-CDM model (AKA concordance model or standard model of cosmology (SMC)).
But you can only determine them from a known luminosity such as for SNe Ia.
Luminosity distances to SNe Ia are key data in determining the Λ-CDM model parameters.
The luminosity distance is determine via distance modulus.
The relation of distance modulus to luminosity distances is here derived:
where luminosity distance r is measured in parsecs (pc), F_10 stands for flux at 10 pc from the source where absolute magnitudes are defined to be measured from, and we use the inverse-square law canceling out the common factors in the derivation to get the expression in terms of r.
The -5 in the distance modulus is annoying. It is a result of using 10 pc rather than 1 pc as the fiducial distance for absolute magnitude. No one ever changes a klutzy convention in astronomy.
The luminosity distance formula is
The following table gives some fiducial distance moduli and corresponding luminosity distances.
__________________________________________________________________________________ Table: Fiducial Distance Moduli and Luminosity Distances __________________________________________________________________________________ μ Luminosity Distance Comment (Mpc) __________________________________________________________________________________ 25 1 Fiducial nearest neighbor intergalactic distance. 30 10 Of order of distance to the Virgo Cluster. 35 100 Nearly the distance to the Coma Cluster. 40 1000 Corresponds to cosmological redshift z ∼ 0.2 45 10**4 Comparable to the diameter of the observable universe. __________________________________________________________________________________
The light curves for the other kinds of supernovae will NOT fit the shape of the fiducial light curve and no matter how you adjust the level of their light curves you will NOT get an accurate luminosity distance, except by chance.
So you will not get high accuracy luminosity distances in general for those SNe Ia, but should get not-so-bad ones.
The SNe Ia in the list which are peculiar may or may NOT have anything close to the normal SN Ia luminosity.
So you may get very bad luminosity distances for them.
Note the Supernova Light Curve Fitting Explorer is just an educational tool, and checking on it's accuracy is part of the education.